escola oe geometria diferencia - tu...
TRANSCRIPT
ESCOLA OE GEOMETRIA DIFERENCIA
UNIVERStDADE ESTAOUAL OE CAMPINAS
21 .25 de Jullho d. 198
• 8trocinada por: CNP' ~~ESP. SBM.
NOTES ON ISOPARAMETR ,I t HYPERSURFACES
DIRK FERUS
In ll'"Orluc lion
We <Ir!" conc,-,rncd "'1 th t.h.: lltud)( 01: hfP(;.t:surfa~i:'r. in a staod.ard
opace 01' C'on:'lll:llnt' c-ut"ViJt:\ll'C: ~lI'iJJ Idean »pUCI2!,. $phcre or" h:,'oor
h<~,l cc· Sp;:H::e-, J\ccord (nli 1:0 I fLino 'iyper('luc
tüOl:!5 .. ".~ unÜI"C!ly c.,t(!rJnlrt,··d u t.h~ ir fies
ami SEl"COItU fund.'!'ll(!n I.,
ll"ft~orHll lnvar; _,n L' aturll.li '
An it E!11l5
e rstQndlng oE tile c
eOnsL.l :i:
Ho'" do hypn sll: pr I nC I. tJiJ.l Cll[ Ifdlur"!J
k lil<c:?
1'h 111 qU..."l Lan ",el!. 50 111,...,\ far hyp{![,slIl" file","!'! 111 eue) l<h:wTI ,,{'ilC€:
y 1...,.... .1-Cl\lila (10) und S('<]re (~7J I" thc JÜHl tldt"ties, At n sam'" Liror.! E, Cilrt"l' solv"(J th~ l'lliperL,vllc c.)(;l:·, In IJ,lth (' ':lIilS
he rl\~ll'lbl:!r q 0' istinC'L jJrlno:;:I1'111 cLlr\!.:Jtue.~s .i" .... L '1\f,," 0,
tll .... hyp",rllurfac"lIl look,,; 11ke " tub.. ,]rOLilld d tüLrllly
gcode!l'-C 5ubSp.:Joce (1...." :JroOlrad an .\LflTlt~ .ül1hBpi)r~r.: in tlw eUI~li,
cl (. ,'" cas... ) , [-1 J.
BI.lt. 11'1 U1C :;pher i ",""i C"'HJ ound tnr GltLiALIOn quj
diHörQJll [1 lUl.l~l, Tli .. orbi of Lsor=l,·trio:lB "'<;:tiJl9
on th" 5pher~, oo'JÜ'u.:.;1 1 CU I:'VQ t !l res ,
LJJ~y ;,aplu'rl t" 1, C.U"!;.l.ln was .'1010 to <';0[1
h Up tu 'J 4 clistiflct
l~o Dhow, that for ~ ~ 3 G rnust. hornC9",neou5,
Rema.-Ir.. Hyper!iurfacC"s 10'1\;11 cönsl:'::lftt principal curvaturu; nrp
charact... ,....l.zed as level hyper,.;urf."\c!2s of functions fn
'<Iflieh thc first a.nd sec;oncl "d1.ffer>"ntJ.al parameter" rl."penIV
nly on the value of the fu"ction its.~lE. They (,c,-, tbd:'efore
callüd. i .';upara,.:Ol tric h!lper;Jurf{we~.
I
2
,[tC'[" "lI!.'!.:",n lha 'ßLlbj",:;~ fell il\t:o ciolivion unI i.! abauL 1970.
IH'OtJ<Jh t it up agaln w1 t.b a survey on tll
nown "eIiLJHt; n:1rl Open guestionI> (14-1. Tule,<lql and T<lh~hilo'lhl C1€·l. .otlc~d th.1l:. thc clu:;,ttl I r."ti",.. ur ho""0'lel~eQl)S hypersLJrfact's
q1v,I."" by IHHan9 and l.,a·...sQI'l [81solved Lha
'I 5,
ud _" nlou,la,' ...
s r Ii dj!ltlf1cl' i:'rlncjpal cur
nccy. ~~ong the homog~neous eKawples,
cu ~ampIQ~ ~~r~ knqwn ~ntil lh~ surprlsin
Id I'ld '('a~eur.h 1. wllt"re 1.....0 In f.i n.L te series 0 f
lon-ho1l10geneOu.s hypen;lJ.rfacE!s 1011 th 9 c ~ "'Ger;, eon, 'Jc~e(! "
:tiJrtinOl frollll thJII; P'"pll I' , !'Iaceller, il iln:tnCll" an,'! ]y
und ,j''ilm'tl'l " mud, l.'lrq"r nwnb·(,r of (9 4) - examples" 'rhoy
,n" CClJl8l:.tllt:t,"11'l in I!l unifL,·<J Wtl'l' ustng ellfford
\!ld wrllt=&n a for,m "ihic:~h ellsily WI-' think,
'~r"l lut.er1:!.s,Un !!'Dm , pilr'l;lOlll.'H' ... cl
·lntto. but Ilrblt.carl1.y ~~pac~ riemannlan
\&nlfold,; wl,ich lun nQ~ 1 !'I om... tri buR tlava "tho same" curvature
.ensc,,; •
. n My lee t.ures 1
IE'ntlonad 4b;:n'u, Olle P cl. js th
~Q~Ü!t o iilpllc AI LeT I!ihQrl~ prelimina.ry
,,-,,;:tJ On On Il.hc ~tJ, tlOIHi 0 Gdaes ~nd Codazzi. ~o bc~in with
ud'" l;.1)e shElpe ope eutor on <w HIIOpf.H"a
We COl"'~ to di,tfcl:oant o"Pl"o"elles:
l.ßotller qeoUl!;!tri ,ne. doscriblnlJ DU <lrour"la
:.beir foeLl t nl[ohJ.;;, und /'In "f'211lltle descrlcl:;.!onl:hat will
: int us irito alfJebTil,
s)'pl,11 no~ 90 .1.llto ~Ün:l-ll~r'!1 P1"oo! [-<1.1 Lbut onl)' 'l "!' 1,2.3,4,6
re [lCl!lgible: il.. useg vel'y I n~-rlc<!tf' eotlollTO!.ogy drgUlllcnt:l whieh
ce beYOlld tohe scope of Lhese lecLIl.reS. !<nd' ! sn!lJ.l not.. ':ja int:o
11 t.hc detillllG of tho CLl.fford Qx;rnrples ei ther. They wi 11 be
'on~1nßd ~n a paper 1n prßparation [71
y notes are partially based on leeture notes by H.Karcher, and
.n Mllnzner (11] .
3
'],h~ e.qLJaLii;m~ cr GAUSS .md COD!\ZZ,I.
11.. main int~ntion of thJ.li prel1fl1lnitTY ,"ction is to fix our
flol>1l1 Of\.
1.1"L" b~ u ::ltIL:tl.mi(old of an (n",,1)-d~ItI"l'Is1onal IIp.1CQ'i:t of
nstant cu ..val"urc. C. By .t ... , ... '> WB ahall defloto 't;lt",
clemannj"ul IIlf!Lrlc Qf P.J lllcl' manifuld" Tha Levl-C1v l t.a Gov", .. i 1I1IL
,k.tlvatlVI:! Dn M will be denoted by Q, thiJ.t 011 1'1 by \l .
I' X and Y <Jro t.olnS<ln t ve-c tor f J.~Ld~ on ~l, t.llen" ",j th t.h[~ U!:H]1l '1
j dl"nt i I icöU.onr,
'V .. y 'V X Y I ~x , 'f) , (1)., ""h Clr.. n (1< , Y ) ! ~ .~ 1'\'-' f 11L11 Vm:: tor f 1. i:! lel. 'rh j!il (le r I fI ,,~: tI ~ ytn"'lllr 1 ,~
billneiir mil~ h [ran, 1I ...: tllhq~t,L to tlla norm,]l alJtlc~ eul h'd t.li'-'
IJl1"O/JJ rJ••O(ia,·~.l"lt<11 r..d·", o[ 1'-1 j,tl .., '>(. Fu~' thleh nUTmll] tipI<! ~
W~~ ob'- 111 S'/l"trlf..:L:r ie (;!nd{ll'l1orrih..i I,I;;m n9unt v(!ctor~
<: S'l' " >:- ~ h (x: ,Y) ,~ .- ( 2)
Thc tCrtSOT" S : .1.. ::> End ( -i ~ "r-d 1103 on.all' Orl!'.l"JtOl"
or t:"('IIIJ ,-,1' o.f "I i f' M"er • NO<'o':: '1 .... jth dis! Inl1u l ..d UnI f'lt"!flYl
[leld ~
h (X, y) i nst.es,cl of ~ Tl \" ,Y) , ~ '>
.::.nd
s 5 t-e~t1 of s'( .
111 this C"S;~ h i'l '~unl- luud o1n S :;lmp" symmfllrl e.lldo
llIorphism fjeld.
l1e covarl d<!rivativC' nt ~,ut!';t.>r
fi. . g'oin intn forrnnl det~jls w
r'cC<loll, chat 1 ll-Slng lhc- ,n"OdI.lC HilI.!' OS " 9U1d I Q')
prin("apl.~. ,0 xample, 'hp, c=ov~l-i[jrH <1 "J,-l ivc "'I' S i5 <k{ined
by
('VZ5)~X := Vz(S,,:!.J - 5",zl,X - 51; '1i' , (J)
whör<; "vz\. 15 tbC! normil.l eovar ianL der 1 vab ve ( 'Qrma 1 c;pn!l'0oun t
of ;;]z1l
4
,. R' illnd b rl80J';'(; of M and r~ _ Thf..!n tt ... tllilt.lO!l Pt Gl\USS nwds
R(X,Y)Z- + Sh(Y,Z)X - ""h(;",2) Yn(}(,YIZ - '"
ct<y,Z>X - ~x,z>yl ,I \ ,( '( , z):< ~ Sh (X , Z) Y (4)
Fbr hyporliu~flll:("'" w:i (-) 18t'cd, uni L nOrmal [lL.,1r"l, ttJ J tl r<!duc... ,. to
Rth.Y)Z c r~ y, Z'> X - .( X ,Z > Y 1 + <S'{,Z>SX - <SX,Z>S'{: (5 )
he equillt 10n cf COOl,ZZJ is
(<;IX S\ Y ('-'yS)l;X, (6 )
~Ild eil" be OIT1J!Ij, t in thellyp,)r:surfa(:i- ease.
r proof cf (G) seeCl ('I) ~ r 1nstance the boot cf R NOlllJ.l.Ll, vol 11.
11 "i Is a hype"~urfac:::e wit.h <\ distin
an
1111 fif"ld, e p.iqC!ovalUO:!l of Sure calied th,' 'a t!Ar .. ~;~
Gf '" Ln 'i _ Jf tney ~ru COnst~nt, "-hell M 1
i aapaf"a4'lBtl'i", '.U1U1T'O,,,' faeJß.
~he ~o loeal unit nOrmal field~ l~bd to op~ositp üig"n~ of the
pr1nctpül curvdtUre!!l. There foro thp. nCOnBl:~ncy" oE t.ho princi
1 curvdl;ures und Mence the no,tio[1 "isoparamctri c" mllkes nS"e al:; tor hypersurfJces .' l~U. a distinquished normal -ld.
5
~s .
Le t M be ,Jn t:Hl(li:II'ßmett"ic hypcrOjuJ~f ,- of the ~+ 1 ) - SPIII:u 1'l .
'IlJe foUowtng cOh.iaorat.ic l~c,iJ we Cdn a5sum~ that
M ha5 q dls~iP9ulsh , 1ft ~ . LA~t A be one of
the principal curvatures of M, and put
-: S - ';\... Id - (7 )
For vector fields X, Y, Z wit.h ~ j;y o we find
< <:J 'i ,'liz > <,~ (<;Ix Y) ,Z ">X
<"lx (~Yl ,Z > - < ('1 "',lY, Z> x
- «Q:.;'~)Z,y> (symmetry of Q' i) "'1-.
<"("li!A)l<' ,Y > (Coclazzil
<QZ(~\()'Y"> + <J}{'1z'<)'Y>
< Vz'< ,~Y > O.
H'.fll~""
!:X >- X ;)nd SY ~ ) Y impl1cs S (Q'x Y) = 7\ ('ii Y) . (8)x
Thtruforc the eiqDnUpJ~U dlatributLönc of S ~re autoparallei in
M. r:uch f'rl.t1c~pal curvatur-,~ A dete:r;minl'f; a .folJ.dtion of ~ by
tO~411i' geodesic: <;ubrnani folds, the ).. -al.ll'vature Z"ClV
Since [or X and Y tanqent to such a leaf we have
Vxy: QxY + hO< ,Y)
QXY+A<X,y">~.
th-C[;O le,~ve~ are?.. -umbillcal in'~with maan curvilture vector
lll'"allel to ~ . So ~ ccnsists of mutuillly orthogonal. families- (~f umhilicil_ submdnifolds of M.
6
I f the [Ullnber 01 flrlnc1 p,1l CU.l;vatures is g.- 2, t.hen
Hit' lac,llly by L.a,kinlj <I "i\.,-llInhiliCu1
lhe mult..ipliclly m, of ).."
tr'-"'!J I..:ltir'l "n orllH)'}önal A -umblll.cal s2
n m ölon lhe fl rRl onc.2
If q "> 2 , thon Lhe $J tuatlOI'l hocornes ntore C'c>rllp1icateo, bac<\usc
for principal curvÜl t~rcs )" * A j the A... -le.. vu& will not i1 . )
ener<'ll bo paral ~131 a lonq the "'1-1eavCJ:.Bul .lf '..c ean gel
contra} ovcr the rotatIon of lhe löavBs around e~ch othet, we
c...:>n aClaJ.n rCConb Lrtll;1: M sLarting frelfl1 a p0111t x and from the
eÜutl1 ces of S at. >:.
We scl!')(:: a prlncil-'a1 cur-, ture A, and a unlt-spe~ g020d"",lc
a ~ -otU"va Lll lear L. We want to d cribe th"
t r- -lc.lvea r- p)... , !lioo') r· Fbr t.hat purpose
to describ th!.< behaviour of 5 - ;\.ld 3100g '!
this ~~nuor fleld satlsfics a frrst oröer
alol) 'f' whic lnvolvQ~ a certain
tensor leId in turn satlsfies a Riccati equal-.1on.
h '.Jquat~ont;, we arrive at CI seconu order
equ.'lUon for 5 - A 1d whieh can be solved
'l>licite1y~
Let 1<: and ~ denote thc ort.hogonü1 projecU.oos onto ker (5 - .:\1d)
emd !m(5 - A rd> n.lnpectlv,cly. Then
Sc Y := -1JVy ~X (9)
dcfines a (2,1)-tennor fie1d on ~.Note that
~ and 1j are parallel along L, and } (10)Sc°~'O
by (8).
L~ Let X be a vector in ker (S - ';\Id). Theo
"7 X (5 - ?- Id) (S - ).. Id). Sc . ( '1)
7
Proot: E;(t.o-nd:( to il V(;l~ I' "Ir' l hI1d in ke r. (S - 'A...ld) , 1111<1
1 ... y b ..' illlY vccLor fii!'ld. Thon
(Q'X~5 >-.Jd) Y (Qy(S - ·".ltl~IX
l7 y (S( - "A:\l - (S - )..10)'V';
(S - "rd) ( - \7y~X)
(5 - ;\. Id) ( - \) 'V y ö< X>
(S - A r.d)· Sc ' .
.1 veet kerlS - 'A1d) , he!fX<l:TM l'
( Vx Clx = '; - + f\, ( 12)
""h,"" I\: 'f : = V 1I ( '\Jy ,x )X .
F'l"oof, 1\1.1 three I'\SOrs 1n (1:>1 vanlS1l nn !<.er (5 - A Id~ =
le 'I be LI; k"," (de. I. \~<: L!Xh.l",d I( and Y ~ p P1114'111' ('de. ' .
vcctor fLl'ldtl on ne i q-hborllooo ur 1:', fll.Jl~h th··
'V;.:X) ( 'V X y ) p = 0 (lad X = d< X, Y ~ '\) Y.
Usi 11' (10) a1 Lim"f1, we [111<\ .t p
("XC)X'l V (S: Y)
'SLo
x
- <J' x 1J 'J,/·
- 'V(\lx\l';)
(R(X,Y)X + \l/TxX I Vv.'; - \Jv. xX ) x 'C
- V R(Y,X)X + 'U 19 .;X) - \J NyVXX +~~'(.qv~
u Sc 2y + 1\ Y - ~ <\Jy'VXX "Y i"> Yi' ...
VYi ~ Y.",here tl1<, Y are ort.horormal veetar fields with 1
But at p '" ~(,.)
:::. 0<-'VY'VXX'Yi> = y.<VXX,Y i ? < V XX" 'V y i >x
\' oThis inishes the pr-oof of Lemma 2.
~
l
8
~. L....JYlI11" ] 1s t["Ue for aflY Lota11y ye'Joth·'1ltJ f",li.,tlon.
In the pr'csent gcolT'etrie situat.ion tne "<tuss oquation yicld
m(.lt"f' införMn,f:lon on fix:
Rx 't = R ( \J Y ,x )x .c X ,x '> (c + ;L~;) 1)" 'f. (13)
No'" let be a unit-5~p~d gcode~lc tuny~nt to A -leaf L.
w wc.1t:e C:'" c~., [l :.i R;; ancl d"notc by ( .. )' eo
derivative a10ng I Me'l (12) rt>nd:;;
2C' = C + R. (14 )
Let 'l bc Im L-oormai vcc:tor neid alOfllJ ~ such tl\
y' -I- cy = O. ( 15)
'hen
o !(" + C' y + Cy'
2 C 2y" + C y + RY - y
't" + RY. f 16 )
Hancc C dc{lnus (and is detcrmiopd by) 3 cn ,'li n ela::;'" 0"
J;:lcu1.JJ fkiclr: .11ong 't' which are in u sense ciFied r'F l{aJ:Jple in [, 1) "adaptcd to tho folt.:Jtion".
We d..,tlnc
A(t) : = ~ (S - )...Idl-1 on 1mago (~( t) ) ( 17)i.. 0 ~(t)
on kt:r (-ae~H)'
Thcn alon'] ~
1) =A.(S-).rd),
and from (10) ,md (11)
o = A' (S - i\ Id) + A (:; - AI d) ,
A' (S -), Id I t A (S -), r d) C
A' (S -)" Id) + C
or
A' + CA o. ( 18)
9
This i !! tensor analoq ,C 1151 .... n" j mpli the fo) 1 ~..... 1.n
"nalog of (16):
A" + RA =0. (19 )
(1) and~ I~~ = 1 we Und
A" + (e '"S)A o. (20)
Usin'J ($ - ":\.10)' ,!\, U we eQoclu<l"
A" + (e + ?hA + 7-1) = 0, (21 )
and arr!ve at the fo11o,~ir"3 c;ombln<1ltion of Lemrnas 1 and 2
Propos '\.- t bt· a tlnj t-spe_'<:l gCLJucsic t.:Jngent t.o
U ,-,nd A cl .. f intJ<I ar; abovc we h"V8
!:or
(c + ;>.,,2)A AU)" + (c + ).,;(j (e + il2)A + ;\0) o P")
ncl {I'JC C + A 2 o
,Ii, " + 1\ 1J ~ o. (22' )
Thesl: diffQnmtinl equ,atir.lns .'In' ~a9,dy tl1tcgr ;<, c)([-.lie-lt..'l'l.
(Note that V iG ~~r~ll~l along L.) ßu~ h deL.r Ine5 S co~pl~t~ly,
und tiwreforc our piröbl!!rn seums t.o be sol ved: we knm.,
rot.ation 0 'ouod L. Thür~ is only ans littl oint
ouL to ~pftjL mu~h of our glorlöu~
viel:ary. We don'L kno... the inill~l ~ondlton~, ':load otlöl.I~h. 1l··:;It!e:;o
A, wh1cn wc ean .:Issume to be ~iv~n ~t our stnrtlnq poInt, W~
neet' ll;.~ dc-ri '1"1 t i. veG WI th res",C'ct to ,111 T..-tanCjent dlrectl.on&.
\\, 'l~"!it: for C >0 ::; unclear, whlch d
.. Sill1oot.h field A on tl10 vpl1erIJ [,. E'ven ....'ü,~5!<: ~ft"r
translaUn<j the i\..i-leaves, 1> 1, <Il011,g 01 ::l.1-1011f, Wl} havr.
to trullslab! Lhe "j-laave&. j> 2 ., <110ng all iche Ä.. 2-ieaves.
ßut for thi< WIe f1o~~d th ~ lnitÜ>1 data tOl:" A = i\ 2 ",10110: t.he
~lhole ),1-h~i f.
)o.~
'{
10
tl..<!vet't.h"l<lSS. proposi 1 10n 3 gives valuable inf(;Hll'I.J'tion
at "''I 4HI now ']oinq to, 0)([,>101 t.
'~h!.! cqua Licns (22), (22') hayc the solutinns
(c + ;...2)A +).1J = cos lctX E + slnlcJ-X t (2) )
A = - t 2 / 2 )., Lr + I.E + E (2J' )
h fl'ol'Ir411el tenlH'lT [leid E, E nlo~~. E.~ t'; • ~ : O.
Ir ;; I- )"l ... 0 '"', sungl i uta eosh ~;:'i't. c. a ge t ru•.ll
:u:Hu t1 cn!l . I If th~ ei'Je~v~lu05 of 6'f."(:
A ).3' )..g
'Oll Lh 1II1.l1
In,' 1I1 2 , m]I m IJ
t~cn he elql~nw\ll""t; (-' ;\ ar- I:l 1/).i-)., , hd
(<: + )c2., A + ).. 'Ij <ir.;
<: + ). ..)~ 1.1 = 0, "1 2 >::- ~. , 19 )."/
wHh th<! sam, lll\J lf. ~pll ·lties. and th~v ~r on5\'011 t c1J ong t· ~OLIl~ r t,lle • n(,) eCPsBarily diGtinc~. bccnuso o1r "1.i IIh1) h... (J I lor an I slnce we ~re t~nlJliy 1nLeT0, "<1
" h On1" 1110 " \ kill' (d{ ) his will causa no cwblc.
the e.j'lenvlJl'JI"~ un t.he (jql1t htmd stde of (2'j) , (<'::I' I
mUEt I2:nt oe t. '1'111,; im .rtiaul.:lr tLi "" 1 n In C;:lse f 0 the trac~ 0 E (;,nd E ) mu:;t bc zero. JI1
CIUiO er + o lt follows ";\ 'co O. In e1l;her GiI~;C: w~· IhlV
. &_,, .0" 4. Fbf" hypersl,Il'hce wj th d1stinc;t.
ClJrvat.ljj ,." ). o( 'lIultipllCity ffi 1 , ... ,~1 in5 g
aspac<.: nstal1t Cl we have
~ c ., LI.; ( 25).?~f1Ij
j .... f\' - A' ° 1 l
j 4-:1f: aU 1.
11
(25) .is tan J-. i~oparIlrnatr~c
hy-persul:'L.a':t!S. }roml\;; hf' oit>tained the cla!lsj f Lcnt~on in the
C~SC5 of non-PO~ltiye C.
Coas l<lpr tlrst l'he Cf",,' Co< 0, say C: = -1. By ch".n<J.Lnq the
norm- ~ld if n~ce~~~ry, Wo mnv c,5urnc tha Bce ['011l ive
;\'.i'" L~t.
)... : = !; l1p { )... l; 0<' ::t, ~ 1] 'JflQ '1:- lnt{J. ; 1< ).. ,~ . J. t 1
"'!'I~lllll"" first lhaL >-.. c:d::;tl3, al\O i.h thore ! f; no im:lph 1 curva,l.lItG· 1n })., ~ ( . Tln,:n
-, I \. i'- ,. \; - 1/>.. ".) - ?o.. ~. - A.
~
1s positive 1:0[" <,11). f >.., e.-:ce[ol poss1 ly ;;1... = 1/>... . Irene." ,J
.n '] > 1, by (25) we hov(: fJ 2, afjd >.., >"'2 = 1.
ASSllITII' n.~:<t thut ?.. ,~:." ~;t::;, lJnd Lil."c there is Cl pr! nci pa ,I (:11"'
valure in ]l-.,\r.T"hc.n al:;,a r-~]A\"t-,,[. andlhlJra 1& n
J 1-, ,... ( .woprim~ipill CuP/l'ltun! in e,1fl therefo'C<: dlJply th"
(11Jove a r<JUII1Cll \. ~() l" ins tead of A
If'.i 110.11", i [ \..t1 j,:; no ::\'1 in 10,11 , thren ,\" exi.slJ;l, dlll
aqil J n "] ~\' ' t" r does ,mt cOIH;filn nny Al' .
Fot' ~ = 0 a lfil,lrh t;il1lpler gUß1il1l"1l wad:", which 1 I.mllfLJ to Ul
C;tdCL". We ue
.Ir 1-1 1s an 150P i 1"1
M nt curv~turc C ~ hl.n the fjUl'\l), or cllst llict
PC1nc1paL curvatun~5 ,ir; g:!O 2. FOr <J=" M U; totDlly umLilic:
and verJ' "'oll kno·....n, i f M it; a s Lal'\a~rd space. fbr qc<? th.e
two di st:i r,H.;t CIH.:vatU'L"r,i!S !.;~tii;fy
- ).1)2 = C, (26) \ ane! ll'H"'lj' ,'1 Ül obtl,Ür,m:! by taklf'l e two curvaturo leaves
(,,.:'h J eh are totally umu.H.LCul Bullm "J folds) thr(1)qh ,) <liven ,"0 Lnt.,
and then (M-lparallel tr~n"Jatlnn qne .. lOri he other. --.............
This solves t:he loeal Cl<l5Sihcat,ion problem for C ~ o. One
ciln show tlla \; cach eonnecLf'd i ,;oparamet:;r:lG llyp,~rslHface i S -In
o~on part of a complotD one, and r those th~ classificatlon
15 the SJmc. In euclldcJD 9pDce thp isoparametrlc hypccsur
faces are (open parts 0.) gph~Ces and hyperplanas (g=1), and
12 13
i"'pll C.",
s"hprica 1 cyLLndact; (<]"'2).
('1'1 - '\'t.~.,) + ('fi - Cfi-~) "E> 0 mod)l Ir c> 0 ~he abov~ arqurnsnts fail. bu~ proposition 4 stil] and
n Isopuri!UJietric hyperDllrf.,ce of the
gi..,,,=s lnt '!sting itiforrnatiQn .1bou~ l'he prjr,cipOll curvab.lra~.
Ifi - 'fi+' =- 'T' i - 1 - q:> 1 mod .... ,o,t curvature 1. Let.M h~vl! the dist.lrw , l'Id I ces mod
Flence the 3'i apl:!l1~llP"l CUI:V Tcl '", <:A2 < . _. <'\ "'1 th corres[JOnditl uidiatant. modT. ~'h!s proves the proposition.mulllj)liciLlell In, •. _.1 "'g' Thon. for j, ndlces /lIod g. we lHIV
= mi .l 2 (27 )mi
ami Ln!, DX,Lsts O((~ '10 ~~C r~;l~ch ~'hi3L- . ~
/...i = cat ( b< + (r-l)~ ). (2a)9
lf 'J io odd r tllen aJ.l multtpliCitlc'- • e<~lJal.
"";>0Proof: PUL Aj : = CO~ ':l' with "Ir > 'f • ., '" > Y\ ChoO!m same A ~ Ai' nd d<lfine A corr.-:sponding.ly. hi.....n b'l (24)
the e1genvalllCs Ol (' + )..2)A , AI) \ ker ('af) are
+ )..; ). i == cot ('i'i -S'j) ""lU, mUlLipl. fI'1 j (29 )
A~ - ),;
E'<tlluatln'J ( 23) fOl t = 0 and ""'i~t.="To- ~ho"".l!s Lhtd
with eQC (,~'i - q'j) also - l~1: (<fi-~j) i~ an eiQ~nvslue,
...hleb tly I:ontinui ty lliU.f;t halTe ehg sa~l!'-' l!lull:lpHattv n,
~'rom
"ir ... '!. _. er,_~~;-'!i.\~ -:. 'f; -''f') .c 1\"" 'f',; - 'f, "' ...
we obtain
c.ot ('Pi - q i+l)""> ..."l>c:ot: ('f i - ~q);> cO~(~:l - P1 b ... > CGI: (Cf, - '1-1)
wit.h Gorrespof)ding mult
1111111..+1 ID i - 1
SLflCC (> nC<lat:.1V'e th 19!J iHI i n
~ t mus t be t.he sm;:, lleGt Oile ß' Hene m1+ 1 = lT1 J.-1 ' mod 9 • ~oreover
GOt. ('Yl - Cf 1+1) ot (T, -o/i-1)
14
,ie eoncCrI,tcute on the nphe.ical case as the ß\O&1. int.en:PtlI.1ng ono.
cut, lllQl)t of tohe following considerat1on::: earry over tu the gene
t"Cll slt.lJl!'tlon_
H be Cl hypcniurLtce of tilc st.lI'lJa[:!:l aphöre Sn+1, and lpt
~ ~ a luÜt norm.,l Huld u]or;.q M. Th('n for mQ;;t: r:-c.11 numb
t the lNip
P I-'> ß;,(p_ ( t ~ (p) )
defil'iE!& 8 paf'l1lLo? flYPI1'rsurf,J'J€ Mt' und the shl}pe ol'E',rutors
S olnd SI;; of M an,l.M t ,[Ice rcl a,tcd Ln u ve,ry simple ",,,"y:
n Gf.';lce::; of S t 'lce 1;l'~rallcJ (even in euc:l tdt"'!1 "pace)
d.LOn']' Olle curvc t >-~ el<p (t l;(p), and th;:; elg"I'V\lölue of
l: cQlrreapt.>Ttdihq to the principal eurvaturu A.. on M is
cot ('!' - t), where A ~ cot er . (30)
flenet:, if M 1(; lsopariJIw:;\;r:ie, so 3r~ the paraLlel hyper~'Jrfaces
t' IsOpiln;U";<,trie hYP"',C!HLJ:fttGes Gome in su-ca lIed i:~f)pa r'l1m« trie
;amihcu.
If l'I hdS principd'J curV'<ltllres ~ 1 = cot ':l'1' .,. , "'g' cot f 9
wlth m~ ltipl iei l;.l,e5 1n 1 , ... , mg , thcn the meun curvature of I~ t at..
tue correSPQnd1nq point is
Ht .1 Z m. cot (tJ'. - t) (31)n 1 }1
I[ eac" Mt (for values t in some open intervall has conlltant
II.n eurvaturE:, thcn () 1) is an analy tj.c lunction of t" th:,
polal'l ~f ~hieh.detlermineuniquely mod,.. the :ri' <lnd hence the
curvatu~es. This yields
. --r~'--~-~" '1. If aU pilrallel hypersurfilces Mt or"'1 for
.iI::iently elos;e to zero h<lve cons~ant mca,n eurvature, then
ve c:on'Star1:. pr j fleipal curv<) t.ures and are lsopararnatr1e_
oeal '!1Hlt of M is the set of singular values of the map
p) _ cxp (t ';;(p»). F):om thc eurvature foliations Jf M
we IIce thaI: '='0 each mi~fold principal curvature Ai of an 1so
(1at:lUl\!Btrio hyperuurfaee there eorresponds an (n - mi)-dimensional
15
SßIOQ'f;;h foeal IllAnj told "I ("\.). and thc hypeJ;;surface is (.100 lly)
a 'tube around M ("i)' Thc correspOlldul1CB of shapc operatol.'s
dl?scrilX~d above r~!'lMlns true on 'ehe Jeal man.ifolds in the fOllowing
$OflSC: p~ I!l rJetonninsEi a p< nt p..~M (Al) and florJllal lIni t
name] e tanqentot"lDal r:eat l1e
t; = Cfi'
Then the tll<1cTl-- v alues the~(~il-~Iwpc operator S1 'Idth respeo
i ar
cot (j'j - <.Pi) jft i, ....here ).. eot (32 )] Tj anJ t:he
:eJ)l'0nd to tho~e 5 at p un al o ng thc normal t eirele.w U!J€ thaI; on tlw ':ocal mal'll 'fold~ af an iSOE>ordllletric hyaco I:he ltles of the Ilhap
GOns tan e: rlt and oE the
~'iven a vectur. Conversely,
'-lIbe.6 arQund i
of
,ruYrfaee a;!<9l1'l d_~~~
ase<'! On the difft>n'nHal equal:ioni carrte-d out in the 11n(.
can!:.ext 0 r:iJ oca.l manl tOl<:!:;. COIL'fpare (~\l) .>no (31).
l.rho Vr.3["u
exp'J lei tu kno....ledge of tht, l!'rincipa I n"·\Ii1tur~s als 9.l U CIi u, ~ very good global picture:
1...0 t}\ !'oe a /'O'rnpac t • ,.", tsd i c>tl'~ram~t.ic hypersurface in
njoJ with un'it nOrmeHeld ~ a.nd elcl'enva] lies
cot. (0( ~ "i 9 ""') of multiplicity m . i
Then
CI-i ~""'1 : p ~ e;<p «(w ~-~". )').(p)
16
is a sUOOlersion Ol M OlllO the focal mllrdfo1c;lt!(:li)' "'hieh is
lhol'efok'~ eooneet<::d. M is ... tubu aroUIIO eae" H (Ai) wit.h the
).1 -leuvtJ-s ,.l:J l(1 - f 1bnJ". .ll ,,"'e l, x pE M, then tl'le nnrfllal
r041L cJ.nrl(~ t ...... eXfl (t~(p» = \cos t'p + (sin t ),~(Pl wiJ1
Lhcreforo meat the '\-le" f th!'ou'Jh p l!Ig .. ,j n at
;.:p (2 (0< + :::t~i l'")~ (p» .
This 1uj,p Li.,."
M(Ai ) =''1(:\i~2) ()) )
wh.'re by conlra!;l with (27) Ur .. ilidiclls are not mo<l g.
.. "I', I f, t.,~l t "(~.)\..~ll ..M
~.",u"
r ll"(,,">1, ~ r,,~,) n ,l ~
t11>]jO "i' r~n
~.,,,nl')( ,,(~,) "('1,) "'" \,"t>,1 j, "",1
"'('.1,~,,-------- "",' "'()\:~ ru),)
'" .~ o~~ 'I'he th orum 1 exponuut Llll lnilD oE M i5 th n of
milD! fol<1:; ami 0 oE M,
nd t COnU){lct ilJlu op
famil y by M 11119 thc wno1Q fipherp.. Each hyper uri".;;e
div id two components. Thc~eforc M has cxactly two
s, and fin+1 ia the union oE two solid tl1bes
cl)"ound ehe tocLlI mani fold,<; "'hieh in LersQct in 'I.
'i'his topol0<1icil1 sI tu," I- ion W.:iS s tu(liad by MUNZNER C 1'2. 1 usln~ cohomology tbeo'y. He ~btlJined
',heol"em fI". ['11.] If M is an i50paramet.r;ic hype.nlurface of Sn+1
with g distinct prine1pal e\lrv;:ltures, then g~{1,2,3,4,6).
(Thc compactness-assumption for M is not needed in the theorem
for reasons to become apparent 1n the followlng sect1on.)
17
We close t11-i 5 seetion wi th a 11 ["~mi'lrk abollt. minimlllity
quuHtlon~ in connecLinn wJth 0[' Jmetric famiLics.
Th., rn~all Ul'vatbllr<: vector '1 oE <J ul.ll11llifold ls chardcl;uriZ('(l
by
n < "1.' ~ > tract: S};
or eac ~. Sine€! 'thc rlght hand siele 15 eOnatanl
for i-. jlo1d~, wo obtain the first assertionCJ\:l.) 01' (11
followi 'lq thOOr<lm. Thc eQcond one follows easily from lJ1).
OCd.l lIluni fo1L1~ of an i.üOpf!t'an je hyperJjl,lrface ubmllni tolds. E«ch l~;opa"am<:tric a1l1ily cont:elns
mlnLmul hvpcr~ur(dc~~.
Note t.ho.t by 3 the minimnllty of thc foeal manifolds imp a,c,<juin C~rti!lln's ('lam.,nlal eql1l1lion (25).
19
18
1-.-:0-.. i .-. I this section, wh~ch very close.ly follo.... " MUN7.NER ('11 1,1 '" ""L....;col:.(a ,. k:"J')n
+ m2~CO~(a "" i_I be an Ü;oparame er ic hypex:iiurfat:e of sn+ 1 wi teh d;J.s'tJ nct pri n l. ". • 01 1 0 - • ii~
= ~1k cot ka - O12k tan ka.opal r.urvatureS
1 F we CtlOosC 'e..:> €. 10,n ( vllJch thatA. '" cot (0( + U.,.), multiplicity m, ,
1 9 L
2~cos<...> 2 L.'le<'''' 1 t: i ! 9. and the shape o(:)'enltor is taken .... ith respect to ~J sin ""
2 Il1 +1l1t un i t normai fiE, ld 1,. Then there H: a slnooth function tnu~ we can oontinue
(a. b) : Sn+ 1;> U --.. IR ~ M , nH >= <':Ol;2 w cot ka 'J. 2w ''.1.\S1T} tun
efined on an open nei9hborhood U of M, such that :22n l.n kll
g ~ cxp ( 0< - a(q»)~(b(q») n ( co~ l~
;:;1n ~a + cot 9a ).
IOd alM "'K. Cl' to an additive connt:.ant a is the oriented
h:;\-ance from "I, and b is the nearest point mal'. Then
611 cos L~2 n ( COl g1l + 910 9a ) .ftgrad an <= 1 (J4 )
Thi!'l P~coml!S 5c.:i.ll J m,.1 "f' i E -"clIubs ti hl t"anu f Or a.
I::. f <= lIcos ga(Hess a)grad a >= 'il d grad a = O.
9ra a
9 sln gi!l 4 aThc mean curvature of the level hypersurfaces of a is given g2COS 9a I~rad aU 2
by [s1 - 9n cos ga 9n cos 2w 20:;0:. 9a «Iless a>
n 'grCld a 1\ H 6a - 9n f - g2 f gn !!!:!-ml 111'+"'2'
or or
Af+g(n+glf 2 Illc.-m. 9 2
nH >= t:. a. ~15 l
The Söl.me 1.1But from (28) and (30) ....e know nH, and if we res·trict ourselves ion is true for odd g, and thc prauE i" quiteGiJolJ _. Th ..:to the slightly more complicated case of even 9 >= 2k, we get :,udtiOn ()4) implles
from those two equatiuns ßg:r:<ld f 11 2
-g'"(, _ f2). -i;;._ ( .... _ a» ()6)
nH <= :Lmicot (" + q Finally we e.xtend f ~s • PosiLive-homogcnaous function F of
..rr-ifi;') dc~ree 9 onto thc1"01 i cot(a + qr open co ne {tg; t > 0, q ~ u i Then, uslngthu eucZt:dcarr di
untial operators instead of the sphericalm _ Lcot(a + 'l:lt:) + m Lcot( a + 9.2;;:) ones, we obtain theg 1 J 9 . 9 fi~5!: claim of ,~:a. .. onl~-i~ Hll
20 21
n+1Let H: be an lsopiu-aJllr;tric hypenmrf/lce 01 S
net princip<ll eurvatures, .:tnd F d"fincd CIS abov.::.
teo, W rex) ~ eXI, ,
l) F s fil>5
2 2 r 2g- 2'J1"ö1d Fn = 9 (37)
c r o -··;1I::>.f = , (38)
IheL(:. C g2Im2-mt)/2 (- 0 Eor ., odd).
: I L) F" i ho n~:atr.iction of '" hOOlogeneous polynQmial 01' daqnj(> q.
::'onverl'121 y , tor eJch hOfliogIHleou,;· polynClmi <11 F 01' c]"'.-yree
'sfler; (]7 ) <tnd' D8> th·e li.:vel hypersurfacl:L of Fls n+ 1
fo~ dO !soparamutric t'amib· w~ ca11 such polynomials iao
pal"a""HI"~C f"J'1··)~i,mll.
l'rQo!l: (i) lS .:t trivial cQnl><!quenee 01' (J6) • (J5) , and the
lÜ[Tl\.!t",S re.lu,l;inq thc eUl"1iclea.r\ ;)nc] thc: spl'lericoll grad and A. 9( u) Pul e ; = F - s r for !;omc real nurnbor ;J. Thlilfl
lq d C- 2 u
2'11-;> v rg-~. (39)+
G C is tlo e.neous. Choose s such U1?l1; bG O. ;rhiz is
pctlsible by (38), and ror odd 9 we hav s = v O.
Frpm (39)
I::>.g 8grad GI,2 = O.
On the other hand, since the tia15 of G are harmonie,
o = 6 9 1\ grad Gft2 2g .Cd U, G) 2 1 1· •
0 c
whenc:.e all [lcl1:ti al darivöt.lveo of G. of order g+l vani5h. Hence
Cl. "nd lhorefoC'c I' Ü; a po.1.ynomial of clegrec g.
(iU.) Is iIi tlioiplc ccmputation using Proposition 7.
h conneeted isoparbmetrie hypc~surface of 5"+1
15 an OP~" part of a compact isoparametric hypersurfaee imbedded
in a "global" isoparametrie family.
According to Theorem 10 (ii) the determination of all isopara
metrie hyperuurfaecs in th~ 5phere i5 an algebraic problem,
thouyh a difficuit one, because (37) is non-linear.
"!"h." I.i '"
~ of sU() d~cOmpODQn ioto a dirü t $um ofttl~ GlJbal']1e1Jra 1'~. 1'" (3), and the ve.etor subepace
"! - {X E"lI ; X = -)( '). UJing <X ,Y,> 1- - tracQ (1(V) <19 an
inner pro~uct on~, this deeompOSition is orthogonal, and
'1SO () aets J..!jotnet.r iC;)lly on t.ht: auclidean 5-space '1 by inner Jllorphisms. Consitier' the orbit "I of
x :.. ~ U-~ Y2' 0 0 ~) E 'S
!'t'l, Xe"" 'oie have
[x, x] 0 <!;=-;:> x = 0,
and Lt 0110'015 that
k l----? Ad (k) x = k x k -1
a COverin'} mup from SO(J) onto 11. HencC' dirn CI "c J, and fI1s a homoql3ncous <I.nd t:hen"fore isoparamet .. ic hypersurf54. ce of
Thc tangenl ,.·pace of M at x is
™x- = [1- , x 1.
Thr.: geodesie in,l1 w.l.th initiai Vüctor [.x ,xl at x is given by(exp t'() x (exp -tX). Th'urefore thü ;;;h"pe operator
wi threspect to a unit normal V~ctor ~ is given by • x
s [X ,x) rX ,n (40)
:111
S '''' -1 (i '! 0)0 J. 0
16 0 0 ~i
a sj '1Ilple computation shows tha t the pr incipal Curvatures are
::: 1/ i3' and O. Hence.M has 9 ~ 3 dis ti nct pr inCipal curvatures,e~ch of multiplicity 1.
AcCOrding to Our considerations on page l' , the orbit through ~ i5 a focal manifold, corresponding to rt '" O. The isotropy group
of ~ is S( 0(2) K 0(1' ). Hence the foca,1 manifold is areal
projective plane SO(3)/S(0(2)xO(1», imbedded as a so-ca11ed
23
22
\Ia i:on ,e slIJ:'face.
'file ab llI'e u:lCample of a homogeneolls hypersurfac.; can be
g~ne[." .in<;<! vary much. InsteAo of (SU() ,SO(3» on<:! ca.n
"tarl: wi th anj' Riulllt'iTIniiln BYJmMItric I?illr Cl f e(;trnpu.ct t e,
1 QlI; fo-.: ~n orb1 t. öf mllxil\'lnl dim(lJHÜOfl o.f thll isot opy
rep('.~sentau.on . n codirnünsion turns ou~ tQ b~
he rank of. tne y/l1ll'letrtc ;;ir. Hance for plI.!.rs of rdflk 2
we get iso~t~mc ces of the Gp"er'l. Th.: pr in
ure u lt.iplici tJes can :Oe cor:nputod
In th.\,s lol'lY one obtölins hOOQgonc:ous
D1pll.:S witll the valucR ot g and (m1.m~l:
q ks
l) a hY1?ler~pher,-,
(k,n-k) M 18 a p~Qduct of two t'ound sphen,!;2
J (1,1),(2.2)1(1'1 i,:; a tube arouml a prgjccUvc plun!e
(4,4),(8,8dover JR, C,]H, cay.
(1,,1(-1 )
(2, 2.k.-l)
4
" any na l;m:1I1 numbf'r
(4, 4.k-l)
(2#2)
(4,5)
(9.6)
(1,0,(2,2)6
re detAJ.led list, glvJng also the 111~ann1an symmetrie paÜs,
ia contgincd in ["., 8 1 ,where you Cim also f.ind details of the
eOßBtr.uel:.ion ouU 1ned lLbove. Morcovcr (" 18 ] conta..1ns the proof of
TbcoJ:"em 12. ["., a 1 ,I: 8 1 Each homogeneou8 (isoparametrilc) hyper
aurface of the sphere i8 an orbit of lohe isotropy representation
of a Riemannian symmetrie pair of rank 2, and henee conta1ned in
the list given in [~&1.
= 4
,\n (n+l) -tupie (p ••. ,P ) of 0, 21)0 l1D rfc endomorphj5m.~ 0
Ü; caUe<l I!I Cl,' rJ 'ol'd öy" tQrrt, !..f
PiP j + PjPi = 2 IijId. (41 )
'e l'lavc' th!! folloIYing
TIJeQ...em lJ.=, [7 J .'J .Lvell (t Cli,ftortl :>ys tem (p0' .. q ('lm) 01'l:IR 21 ,
liuch thaI" m. : = m iHl~ m., : - 1 - m - 1 ar~ tll'lSi tive, eh•.
xl ,X .4 , Z-"'-P Je x::' ~ .. 1 • (42)
15 ;J 18o,pllJ:'<llIIi3trlc fUnetion detlning an
"'1 .m.l.ly9: =4, andmulUoliCitie!l r/0 ,1I..,1.
1 !'rOOf:
grad F = 4 <x,x,>xx 8 <1> .x) Ki Ig x F~2
, 6 <x , x > 3 + 6 ~ L < P . x , x'> < p , x. x) < P , x , P . l\ '> 1 ) 1)
64 L<"PiX.x,> 2 <x,x >
16 <'x,x> J,
ibecause <' Pix,Pjx > = <"PjPix,x> <x, x > Si j' Not thatPjPi 1s Skcw-S~YIllt:i.lltci tor .V j.
furtiJermor
ÄxF 4(21+2) <X,X> _ 2 ,L(2 <"P x,x)6(P x,x).i 1
+ 2[lqrad <P1 X,X),2)
= 8 (m2-rn,) <x,x> .
The aS5erU,on follows from Theorem. 10.
24
We Dhull now study the following questions:
How m~ny Clifford systems are there?
'1'0 what e.xtent is a Ciifford system det,ermlned by the Indueed
I~oparumdtric family?
Which Clifford examples of Isoparametrie hypersurfaees are
homoqeJ'leous?
2beg).ll with the study of Clifford systems. From Pi Id foll.ellls that thc: elgenvalues of Pi are :!:'. From
o , 1 ~ j. it follows that P interehanges thej (Pil of Pi' whence dim E+(P ) = dirn E_(P } - 1.i 1
er J'" 2 we-have Po(P,P j ) = (P,PjlP ' Therefore we can o defjn
Ei ;~ P,Pi+l!E+(P ) ;o E+ (Pol - E,+ (Po),
or 1. f1J-,. Th,",n the Ei are s"'ew-symmetric and satisfy
l~j + Ej B1 = - 2 dijId. (42 )
In other words, the Ei giv0 an orthogonal representation of
C1iffot"a alljobra C _ on tl'11J l-api,lee E+(p )' Conver~aly,m 1 o givQn 5kew-5y~~tric E 1 •..•• E -1 onD,I. we define symmetr~c
21 m po.··· 'Pm on:IR by
Po(x,y) ;= (x.-y). P,(x..y) ;= (y.x), Pl+1(x,y):= (E y·,-E x).t 1
'fhen we obtain a Cli.fford system.
o Clifford systems (P •...• p ) and (Q •..•• Q ) on:IR21 are o m 0 m
ealled algeb'r'aiaally equivalent, if I':hoy are conjugate under
an orthogonal tr·ansformat.ion A Of:IR 21 ; Qi = Ap,A t , J.
Givcn two Clilford systems (P •...• p I on:IR21 and (Q •...• 0 )2k 0 m 0 . m
on:IR ,then P.l!lQ1; (x.y) I--> (Pix,QiY) deflnes a Cliftord 2(1+il h d"f dsystem on:IR • t e Lr'ect 8Um 0 the P-system an. 0-system.
A Clifford system. whieh cannot be written as a non-trivial
direet sum (up to algebraic equivalenee) w111 be called irreducible.
Obviously each Cltfford system is the d1rect sum of irreducible
ones. and the latter are obtained 1n the indicated way from
irreduelble representations of Clifford algebras.
2
Now from t.he Clifford ropcQI;Qntat1on thool.'Y. ilOIl foc (Jxl1tl1!,>l"
l 9 J. we ob·tlLln t.1..., f0110w1ng facts:
21(i) TncJ:'e aJ;"e lcl:'eductble Cliffo.rd syst. J[l15 (F\, ..... P ) an mmfor,and only for the f0110wing v~l~UG of ~ ~nd 1 = ~(m):
,m 2 3 4 5 6 7 8 m + 86 6 • ~ •
S (m) 1 1 2 4 4 II 8 8 8 ..... '6 [;(m) (0)
(ri) Fbr m$-O mod 4 t.here is exactly one .irroduciblo "ystelll
up to a1']ebraic ·equivalence, for m:= 0 möd .. th\j~(! l1<1"u 1:.wo.
whieh can bo di&tin']ui~htlc1 d~ follows; Cor: m 2. 0 IIIoc1 ~ . 1:n:edu
cibl!:: .,yst (Po,'" .,Pml Jnd (Qo" ... Qm) are a1 braicaHy
nd only 1
tr4ce 'Po" .P ) = 1:l',ace (°0 " ·Qm) = + 2 J(m).m
It toll ws tlu~t for m.$ 0 I1';Od 4 and kd'"(m) there i5 only
oner l <) raie ,equivalenee class of Clifford systems, for '" ~ 0
mod 4 ere a.re lc+l.
We now CU.lIH! to our Se.cond question: To what cKtcn S th
Clifford system determined by the in~uced illopanH r1.e family,
or rathor by the eongruence cl ass of that famlly?
It follows 1mmediately from
t-<: AP1AtX.X '> = <P i At X.1iI. X>
that al.gebraieally equivaLent systems induce eongruent famil1es.
But the eonverse 1s not tru.e. To see this. let (O(ij) be an
orthogonal (m+')-matr1x, and let (Po""P ) be a Clifford system.mPut
m Q.• L::"".P1J 1.1"'0 ~~
A straight-foreward computat10n shows that (Qo •... 'Qm) is
again a Clifford system, and induees the same 1soparametric
funetion as (P'o •.•. ,Pm)' But talc1ng (C(1j) = -Id glV~ alge
bJ:"aically inecuivalent sy;tems for m a 0 mod 4, 1. 8(m}.
A9 a eonscquelee of these considerations the isoparametr1e
funetion depends only on spanlPo •...• Pm) 1n tbe spacQ of
symme·tric endomorphisffis, afld each orthonorma1 basis of this
span is a C'lifford system. \oie eall the un1t sphere in span{p " ,Pm)o
the Chffor'd sphere de·tcrmlned by Po .... ,P and denote 1t bym
2726
Now tor ax ~ the veC'tors Pox, ... ,Pmx are orthonormal,IP •... ,P ). We are thus lpd to dufin : 'l'wo C~i.ffocd. ~~YLi tum:;;) o m 21 wheneeP., .. ,Pm) and (°0 " -- ,Qm) irl1R "" f113()IIT~tl'io1al~& LoI'~a~R1lt:
titers exis ts an or thogonal t.ran~ fo~m tiorJ A of JlR2 stich M ... {XGSO+l.; exisl:.$ PEL(Po".,P ) S.t. Px = x mhJt tne two Clif(or~ sphcres ßre conj t\J undCr A:
(44) L'{Po, .. ·,P ) = A~(Qo, .... Qm)At.•
mSince any on..ho90nal l', Q 0;, L:(P ,··· .P ) anticoJM1ute, the P in o mout" ~.rlier rem J<s the i.rceducible Cl.iCJ'ord y s tWTlB \me (44) is ul'lique).'f dpternÜncd by x. Hemco M_ is fo11ated by the
a 0 r-o(l -\ .are y ~auiv~lenL: 1ust rupL.1cc on·~ . nteuiecUon of Sn"" lOith t (+')-eigenspace~ E+(P),
i by its "eg (lne ca" aas 11 •. "0,,", th .. ,'" ~(P _ , ...• Pu)' We sha 11 show that ~his fQliatiöfi can be
char~cteri~Qd geometr~~ally: thö tangent Space of the lnafI trace (Po" .1":,;) t hrQuqh X~ M _ 1s sparmoo by thc kernel!i of thc non-zero shape
lIndoc geome Ll"ic: '-'';lw valanc8, and tl'l, ,rutoce t.nex-e apera~on: 01' .l!1_ .J.t x. Dut:. I;ho t'oliation determlne~ the E+ (P) ,
tr.lc u.l vah",.::t:: eln.löses {] nd thersfore 2:;(Po,·· .. ,Pm)'
lth Iß 4 . Wu have to cOlllpl,ILe Uw ~htl1pe operat.Qr!l oE M_. But t'l_ GOns,i.<;ts
of l!linCjul.u- points or F. lind the dtfect computat.i.on of lts1e congrUCnCoc ~lalis of the j tJ Op D..1' iJntet ri fa 1':1 I li dUfJcmls only gcolaetric.;.ill di!ta h. $omewnat awfulJ. \oie therofore .first compute) the geom~trlo equivalenc~ ~la.~, dn the aonVer:H:' (lJ' thls
hem for th~ :Iimpler M+ tnst,ead of M_, aTlo than use the faC't,; no'4 tcue in "r.'!Ost" CUli
tnat the eigönspdCeG of the shape Q~er~tors are parallel alongn+l'1 Let "l bo an i.B{]pal;E1I'Ii~.L!"lr.: IWr.cQrs [a.;:e of 5 , normal great circlcs of our family, and the change of eJgcn
""it.h lIIulL.iplJ<:itlils m, ~ ID 2 . axists a v.::I.lues is expl.ic:tt.ely known. M+ 1s sinlpler to handle tlUlnM_,
l.iffpnl SYlFtOAl (Po,·.·. ,Pm) inducing [~wlth m m" tlten hceallSc
:U'o,.··,Fml 1:5 uniquely l!Iüb.'rfll1.ned by r'l 1~'" a OIl'16'tl'Ü:! "'a M+ = { y'" Sn+' <P y,y> =... = <:1' y,y,>= 01
o m r~ 1" follows from (43), that for CHfton:! eXdmples
~dm~ts a set of m+1 independent defining equation9., " Tn "k.l(m) - m - , exci1pt for f.i.nit,:ely müny (natnely2 t) excoptlons. Therofore the assumption m ~ m, ~ m
2 1s At Y'M+
lt vary restrictive. We 6hall come-bacK later to the except~o-.1.. H.. = span(P y, ... P y) (45)
11 CA8CS. Note moreOver, that the congruencc clasE of L:(Po""'P ) Y o:n m
; alt'eady dot...u:11Iined by .., and I, unless m!:! 0 mod 4. But Tf.. = ~ )(; <:)( ,y'> = <)( ,poY'> ... =tK. ,PmY) 0] (46)
'I!!!n in the c,a"" m+-O m·)d 4 the follo",ing proof flive~ ::>{p;P,y; i/jr .
.CD <.l,eometric d"'5cription of '2:(P " _., Pm) . • J o
'Dof of Theorem 14: Let F be the isoparametric function deter The shape ('>,orator Si+ wi th respect to the normal field
Y,~PiY i9 easily eomputed:ned by M. Then the two focal manl folds of ~ are
+M+ := F- «(:!:1\)1 o for X = P, P . y, j " i (47)Sl 1. J
Si+
X - P iX for XE Ty"l,<x,pipjy"'> = 0 for all i,j (48)are interested in
M fxG 5 0 +'. -L<:PiX,x,>2 ,1. "1-=-.
28
N@.." 'liven x<; 11 'Iod, say, Pox XI put
N(x):= {"'l~E_(Poli '::l,P, x )="'=<l'P x) =01. m
ThiB "'puc hil5 c!" l]nsion I-m 1"2+ 1, <J '1ll.l for '1. ~ N (x I, ~ '1 n = 1,
W~ have
y := (x-'l.JI TI E: L1
poY (x+'!) I D E: .LI'+
sp<Jn(PoY •... ,Pm) = spal1(p (x-1.)"" .FIIl(x-'l)) (491LI'+ o
'er Sf 5DilI1Cf' P Y •••.• P P YI -Si0il.J1(P, (x+"'\.) •.•. ) (50)Ö' O 1 o m
E_(S~l (XE E+(P I; (X .y) = <X ,P y> = ... = <"X ,Ly"> ~. 0 T'0 0 nl
= 1. X ~ E-t(P ); (X,;c> = (X ,PlI'> =... = <X '!'lT'I'l> = 0] (51)o
+ E+(Sol 1X E E _ (f1 0); <" X ,y '> = <x , poy > = •.. = (X , PJHY >= 0 1
t X c. E_1c Po); <X • "'L> - (x ,r 1 '>C) = ... = (X • P1Il"X. >.. 0 ~ . (52)
;>:;1[19 the l\or1llll1 qre",t cll"elu t .......-(cos t)y + (sin t)PoY ~n
t;.r<ln91.nc t.he datu to X" (cos ',/4Iy + (sin ;;:/4)Po"l we ol.>t.:l;n
t~_(Sl) MI (P, (x-1.) •..• ,1l (x-1'[I) (49' )nl
E... (S:t) SpHll(P l (x+"!), .•• 'Pm(X+'l») (50' 1
5ke 1 i X E+ ( Po); <X ,x> = <x ,P1\') =... '" (X, • P(fil:) " 0 1(51')
J..M f !< ~ E_(P ); <x,x")=<x,P x") <x ,Pmx> = 0)J. ~"; 2 ' ) x - O 1
r!cllce
T "_ = Sr-I<l.n(P l x, ..• ,Pmx) (fJ SPlJ.f1(P 11'·" ,P l) ~ ker Sl (SI)X m
~pan {ker s\; 1 I' o} '" (nspan (PIx, .. .,Pmx) & np.:tn (1"1· .. · .Pml )\,t. .L
= lspan(p\x •... 'Pel'lffi nnPIJn(p 1"" ,Pml))l.
29
sinc~ P 1x •... ,Pmx € E_IPcl. P" ••..• P l ~ E+(Pol.m Now
o (551(\ SPilfl(P l1"" 'PI111.1 ,,\~CI
'j'hi.s Cän be seen 110./Iä: lE w d 0 f lJ!!~t':on(P11:")'
then (ur uvery '"'l F 0 in cli'1_ the would exil5t
P~sp&n(Pl""'Pml such hiJ.
P'1 .. u
or
2 7 I1 Pli, '" 1'-'1 = f'lJ.
But th"n the 1 \ C1pa~ I011p
spa rl( P1 ' ... ,Pm) • m2 .L , P I--<> ['>U
.iould have rllnk .t~r or qua l to dim 1/'_ 1"2+ 1 ,
contradl~tln9 m ~ 111 2 ,
FrCJlIL (54) and (55) vltl hav
span {kor 51"; 1. I 0J (span,(P x, ... ,Pmx»).J...1
the ot-tnog-ona-l cou.pll!lI'cn t b-e ;.11<;) tllk.en in 'I'/, _. But this ortho
... 00,,1 compler.'l~l1t cont.a.l.ns o"ly vucto["s j Jl I-:+(P ) , see (53) ... nd o
,(51'), nnd has dimcn~ion .ffil 4 m - lll, = m 4 m = 1 - 1.2 l 2
Therefore
E+ (Po) = o;punV,er s~; 0 I' llO- 1 x'1_ 1e::> :rn x ,
d.!fI Fhu tb",orem follo",:;.
,.. ' (54)
30
r he CQr~iI1 and the results on CH.fford syst.Elms menl::jul'led
'ar _ 0 mod ~ 'hcre ürr' li1 + inCQl1gru c'nt
iSOPiH ic fami.lie!'! on Ud n 2k O(GI) -~P<lC
Tiu' only 5:.1hll..~ <:,.xcuptioru uxe m 4, k ~ 2, an m~8,k~2,
1 t ('in\ ho....eve·- shown, th8t in tba-Ciluse ~or these
'B(;<;!S tao qllivalence elassell of CUr(ö
~'st(olll& leaö to inc.;mgruton rui lllil:S, ~'." \1•• (;011.100'·') t ..<\.<o~ p. ~~ {( .
5e51cl~R there
cl (l)!<lAlples
bß ~on9~ucnt?
th" problem af Lhe
\.lOOn (;0louch
is a r<.\',IcJ. probh~m
not w1.th lI!!ultl
Ueltolf!'ll W~
jl15 1'5. Since tbis
, >!1l - lcrl:gLllarity m:curs ol11y fOt' fömall dimaTlll.tOn.. , 2 ,o/~ ~ irBL pl::~Eien.. ~ 1 L~t ol' th.~ 10"",-dimensional C11 Cford o~p le ...
lIal:e (m , ,m2', (~)" •. IllC<lno tllllt tll!!l:e are two, L:hre,e, ...
guoll1ctl:: ic L'quivalence c l .. sses wi th ",,!Al tlpllcity (m 1 , m ) . 2
(2,1 ) (3,4) (!.t2)
(5,2)
(5,10, (1,81 (!l2)
(2,] ) (),8) (.!L2)
(1,2) (2,5) (), 12) (~)
(1,3) (2,7)
'1, 1 )
Hsno:::e m,>lII1jl only f.or.' (2,1),(5,2),{6,1),4!L21.(9,6),(!.L.l).
1'hr: rlllD.llJ.t!t. with th'12 first three multiplicitlc:> e,1n bo shown
~n QP COnql:Ucnt to (1,2), (2,5), (1,6) ~espectively. whilc (9,6)
ocr the tWQ (B.I)~ ~rc not congrueht wi~h (6,9) or (7,8). Finall
one of '''he l.wo (4,))5 (thc IndCl'filiite One with trace Po" 'P 4 ~ 0')
is con91::u!!Int 1:0 t.he (3,4), wlÜla t.h~ o-tl'ier 1s not,lu \ok.I*"._'~ ...~&-.
rhHore l:urn1ng ~o our th.lru question r ....a.nt tv stop btiefly to
.lscuss a ve:ry fasc.inating eonscqucnce of the abovQ. Noto l;.bat
ow: Cliff'ord hypersurfaees hav(, <Jt least throll princ1pal curvatures
different from zero, and are therefore rigid in the· s,phere •
ß .....-t if we take two ineongruent famiUes loIith ehe same multi
pllcit~e., and from each we ehoOBe a hypersurface, such that tbese
t.wo have the same p~ineipal eurvatures (possible by (28), (30)),
then on bot.h hypersurfaees the shi.\pe operator and (eguntlon of
Gilll:J~) the curvature opel:'ator behave pointwise the S,'l,rr>C.
31
1'0 m"kc this pree ;,se, let us def i ne: \~"o r lcmanni iln matÜ, folds
M and Li' have the same eUJ:vatu.rc tens-'.t: (\ X 6 M' l"lnd x· ~ M r ,
if l;here exists an isometJ:Y j: 1';1 --, 'I'x ,M' such that for
t.he respeetive curvature tensoru we have
jR(X,Y)Z R' (jX ,jY)j'
Then we hAVÜ t.tl" following con~e4u,-,nce of Theorem 14:
Corollm:y ).5. For m=:O mod 4, and any natur.:ll number k, t,hcr".
cxi,,; t [k/2] +' non-i:;oIQQtr ie eompi,lct J:' i.'I111innÜm nlanifo Ids w1 th
t;1)Q :H!me curv<lture tep!'lor (in <:u'ly two pol nLs of any bm of them).
The ditflOTl;,;ion of these lTI[lnifoids 1s 2ke5'(m) - 2.
We now turn to tho ~llrd guestion: Whieh Clifford examples are
hORlO9UnCOU&? Obviously, mosc are flot: look :lt the mult.ipliciti~""
Bu't Wu can exte.nd ou.r quc"t.l.on, and also a:;k for 10cßl geomeLri.e
inv<~.I"L.nt" whlch prove ir'homos,-,m.:ou!'! examples Lo oe inholl!o<renQou$,
Let Po""'P bo 11 C.1ifford system on1R21
w.ith m~' m ~ 3,m 1 "1'2 =- 1 - m - 1 > 0, .,nd cons.l,der thC' foeell mi.lnlfold M+.
L~t Nt he I:h,e sel:: 01' a111 points yE: 1'1+ such that thel:e are
orthcnorroal vee~.()rs "1.1 '12 €. l.yl1+ such th.,t
dim ( ker .S"b ker S ) > 1." 12 The1l we have
l.emma 16. N+ is the set of all y€ l'\+ for whieh there are
orthonormal Qo •... ,Q3 .in span (Po"" ,Pm) wit;h
'° 0 , • ·Q3 Y ~ y •
Proof: Fl'J r'S t let Y" Not' By ri'\!?laCin~ 'o,.o.,'m by another
ortnonormal basis of thcir span we fl~y .1!.~hmo that
ker Sp "ker S,p l Y
span ~ PoP i y; il'O 1" span{p P . if'l1o i'oY
has dimensiol' gt:',,,,ater th In 1, compan;: (45), (47). Hence in this
tcrseetion thece 1n a J'oit. vector u orthogonal to PoP 1y.
Tht)o U = f'0(,2 Y = P,Q3 Y• with Q2,Q3 E span(Po""'Pm),
J2
<: Po,Q2 > = <P 1 ,Q] > = O. Now fOr" ,xl = 1 the 111-.1.(> Q ~ 9::'
3 l;;O~teLrlo::, ...,hul1Ice Qo: PO' Ql .' P1' Q2' 0) !!Ir<! clI:thonnr 1,
Qo" ·"Q]Y = - Qo0201 Q3Y - Q0020002Y = y.
CCll'lV.,·t'401y. 1 0"" ,Pm) be ol:~tlonQl"m~~, °0 "" ,OJ 00
" "Q)y = y. 'l'nen QoQ,Y il[lÖ y = 01 0 3 Y D.:t'u indep':lllr"tlt
v,ictors In th int.er"cctj.on 0 'rneis of 1:11" sbape Opo>l"41
or,' cOl:resPQnl'Jing to the nornl:lL eh lionG QoY' Q, y.
fII~~SIW!>OlI!? 9 ~ 3m· < m + 9, <inu in cn~cl 2
I'a' " • P,t f- 'tId. 'rh~n
\i.1 F N F M+,
tl('rllCC the isop,J!'"örn,"tric fAnltly io;, lnh"!!! Oll
'1:00fo PUt , :; Po" ··1'3' Thon P III gYIIlml'~~'i", <tnd
Ith eLle P1' 0 ~ i';; ~i tld COnlIl1U \ t;~ wl. th iJny cHI" p
k1 (p) ("I, _TlPtlt E~tr .. Fnr 1< ~ _ + ) wr" Ii<1Ve
2 . 2f'(:':') <x, x> 2 <Pi'" X)
llO"
hiJv.. IC {P) c: 7
~~ + FDr y cur asaltlllptio 1;>4 .i.s indefird.fee I E. (P), ....!Iew·
<l'~x,x > = 01 h.<l!l dLmdtlfJion 1
{l system P1" . " Pm on R+ (li'1 whos
haE dimension 1-~~2, and 16 jusc E+(P) ~ U+,
Th\lft. in thre CllG"lS N+~E'I(P)(\M+ Ir/!.
On Ute OCJ nöt too ilard to Sh0W hat P " Qo" ,°3 or ~ny other octhöno~al DOsis 00""'03 CI span (Po'" ,PJI,
oi!.ncc the Grae;:9l1lgbn.Lo.n G~ (span (Po"'" l'm) hl1..G d1llli~n.sion 4 (171-3),
the aj~onslon of N+ in ~tmCJet 4 (1Th-3l + dlm (ll)" Mol' ~
m1 + m2 - 9. If we eo~pare to dim"M+ = m1 :lmm' """lme
4m 1 + m2 - 9 < m1 + ,;!"'2
(ami on1'l if)
3111 1 < m2 + 9.
Thj s provll:; Tl1eol"<1{n 17.
3J
oe exceptions not covurcd by t hL'(ltO!n o.r~:
Al J:l 1!O 2, ('\,41<.-'1) "nd 1'0" .P~ :tI'i, (5,2), (6,1), (9,6)1
B) (4,3) and Po"'P" t ~ld, (6.9l, (7,.!!), (~l, (B,l!i), (10,2l),
Tth"J c:a!les Al are IlOlDOqeneouB. One pos;;:1.ble \Oi'<:lY of proving thls
1.5 to uso e.xpJ lelte Cllfford systems (ob""ined from reo.l division
alqebI:3s), amI then c-:onstruct suffici.e.nlly many lsolllctries 1"'4vln~
inVllrll1nt, ....~ \l..o le~ ..""'~~ ·~.c.\<o"""
ar •.: InliOllloqeneous . 'rhis cal1 be
UlP c"ses (8,151, (10,21)
• llöopa.raroet.e i
Olhown ~iff1ili1' LO Tll~orem 1 7 , 0n 1y in
I,U~ "011' I; hove euch a pToof. Bur. ttlei'mu1 t_lpliciti.es do not oceur
111 'l-he List of 11Oal0'1Elneou.1i aKRf11pies"
sideg lhe ho~ogenooua ~x~uapl~ thl'lre "'era twö Inho/llQqnneti1L
(7,flk) kno,./tJ b,"fore, [~S,1~]
1;,> 5~rjes of multlpllclty (J,4k) on
'I'I1use se.rlsli coincld n
usinq axplicLLe CLiffcLduhlplt~i
,,,ys.telnS, and check condUioM (A),(B1.~ LoIq. rf' rlQL Cli Fford ar'"hu ~1. t'"
(2,2) and (4,5),11.; hOlf1I:)<l"heo~ 'xa!"IIples
~
34
2 In !chi:; section we discU5S some of tne quelltions concenunq
Cl i fford exa.mples usi '1g explici tc reprl!sel'lta~tor,:;,
Lot '1,i 2.i 3 be the imaginary units of thc quaternions~.
Define
E ": IHn ~ lH n ,) u ----- iju (56)
l<) be left-multit>11catlon by i Then the 5 art; skf>w-symmutric.j . j
'sat:isfy (42), Lln~ hc.nce define"a, Cli(ford ~'1stern I'~n) , ... ,p~n) 4non lH n $lH n =m2 .
Pyt Cl := 1, c. := i. l' 20; j .. 4. Then the lsöpar,amel:.ric ] ]- (n) (0)
tllneti';m aS!30clated with Po , ... , P 4' is
2F(u,v) = { I u. + ~v~2)2 - 2{(\ILln 2 - Ivn 2 )2 + 4~<u,Clv>2 J
(S7)
wj.ll're U,V 10 m . :::l
n If n=l t.hen F' -1, and m
2 = 4-4-1<: 0.
'lÜg docs not g1vt! an isoparametrie Clmlly. fbr n:> 1 howeve r we flot one, and wo conccntrate on the !;;ClSe n:
(2 )Put Pi := Pi Then
Itrace po .. ·P41 = /trace (-Id)1 = 16,
and the family corresponding to (P , .. "P 1 has rnultiplicitie~ o 4
(4,8-4-1) = (4,3).
NOw F is 1nVilriant und er the following iSQl:nett'ics
1(cos s) Po. (5in s) P, ; s f'lR 1 (58)
{ Id E!l 0( Id ; o(flH, 10( 1= 1 I (59 )
f A lEl A ; A € Sp(2) 1 (60)
The invdriance 1s trivial for (60) and easily verificd for (58).
For (59) note that the last surn in (57) is t.hc square of
Ivl-times the length of the orthogonal projection of U onto
liv = oo:IHv.
35
A point (u,v) € S15 c IH 2 E9 JH2 can be roov('d by an iSot1\Q
try of type (58) into a point (u',v') with ~u'U2 = nv·,2 = 1/2.
USing (60) we m"y therefore assume th,).l (u,v) = (l/{'I,o,v 1 ,v ),
By (59) we can moreovcr have v l < IR, "1 ~ 0, and, Llsin~1 (60)
~ga1~ wc see that (u,v) 1s equiva!ont with a point
(ö,::;) = (l,O,cos t,sin tl!12 o I! t 0( '/2.
But then F(u,v) f(u,::;) = co" 2t, 'rherl~f'ore \;h,-, isornel;r le::.
(58) - (60) gel'lccnte a group ~ctihq tcansitively on the
leve 1 hypc:~&u.rf'lCC5 oE the iuncLiQn: ehe j'a•.,i tt, 1:3 ""mQ 17,m"oI/ß.
(Note ttl1.lt th!s can be pl·oved similarly for drbitrÖlrj" n as
well as for lR or C i[l.Ble,,() of rn.)
On t.he other hand, i ~ wo omJ, t Po '.;Je l,'lt:t a Cl i Pford By" tern 2 2 1&. cl
P 1 ' .•. '''4 on 1\1 $ 1f.! = IR whu:h il'l UC~li a !urmtlon
2 2 2." 2F (u, v J = (I u ft + Qv ~) - 8 <:-. .c: Ll , civ';> ,
and r:I tamtly of mu.lt:iplicities (3,4). ConiHder thu focal,
lIlanifoldM+ Clf thls famlly. Obviolll;!y y ,= (l,O,O,l)/{2
15 a point of 11+, and aot this point .we hav~ by (47)
;.:.( ..... s+ span t PjPiy : j ft i ) P1. y
span f (Cj~l,O,O'~jCi) j f Il ' whencL:
+ .n kcr sp.y f 01. >. 1.
On the other h~nd for z:= (l,O,O,O)E M+
+ ker Sp.z span {(c;iGl'O.O,O) ; j f 1 1
1
{(x,O,O,O) ; -, -x )
1S independent of i, [md hencc t.he l<ernQls intor:>öc:t 1n
3-dimensional space. Therefore' 1'1+' and hunce thc Ifarnily is
not, hC}1nogdr/.(!()U{3.
Jb
Finllll~ .... e C<ltl CO the abo:l\l~ COnSf;nlc::tlon f Caylay
numbers Cay ~,"'5t"'Qd of I-h~ l'junterniQns. :ror l'i=l .... e obtain {1 6 . Cilfford Gystern ~~, ,PB on Caye Cüy =
Th~n ~~, ... ,P4 and P;' Pa ~re t ....o CIJfford aY!ltelll~ 1eading
to fam.111",~ ol muIt.1t>lJciU,es (4,3) and (3,4) espoetlvelv.
Slnl
z <p~ IU,v), (1I,v»2 ('lU\;2 -IVlI2)2 + 4 L.Cu,civ>2 i,L
(, u ~ 2 _ liv ,2) 2 + 4 IUU 2 Iv 1)2
ill~U2 +IIVIl21?'
IR 16wo üee. 1:11 or K - 11.1, vJ 8~.
x~' - ;/ ~.criJ(·x",>2 _(gxn 4 - 2 L .,' )2)'-l x ,x, . ].=-. h·c
Hrnr;{. the 14,Jf-famil;,' ~nd the (3,q)-f~ffijty coLncide.
By t.be alejellt'lliC resu1tG c1ted eh~11er{P1"" ,1'4)IllU!;t }j(
~'11.1.i"031enL w~th tp;,.·., I'B}. fl"I{t"f'ci:;e: CO/lsl;ruot an expliclte
eql.ltv<Jilence!) ay 'contras t, .tl.Lnce ,,~ 1'4 antlc;Q1r~nutl!!a. w.il:h
5' Jtll trace mus!; b,' zero, and(p~, ,P4) 15 fl()/: geoll1etrically
eqll.iva1e"t wtth(l'''''''''P4)above.
Tc sum up uuc resullS: 'I'here ar0 t..... o Cl~fford eKalflples .... i.th
tttulUp1Jcit.l.O& {.1,3} or (3,4). Th!:' def.inil;e (4,3) 1.5 hOllJogeneous
~e ehe d~rlnlLe (4,4k-1)s), ....hile the lndoEinite (4,3)
colneiclus .... iEh thc (3,4), and is inhomogeneous.
1', 1 n~rnil.rk.
We kno.... more upaut the geometry and top010gy of the C1ifford
e1<ampies than dl[]ClIssed in theS·e l'Iotcs, see [7]. Thc central
probiem ho....ever r,'III" Lns open: the classlfication of isopara
metrie fami1i.es .... i.Ll; '1=4 and g=6.
37
~'-Ir02glW
Carti'll1, E:. t'ami llll[; Cle !'I aceti lt>i)f!.-Il:.:lIlIbttiqUii:'fi dalH.
oTlstl1nto, 1\I~nali. di "I •• l.22.,
2 rtlln, Il:. Sur de:!l fS]I\ i lIes l'"e.ll\lI('guable~ cl' hypenwF ldcn3B
I ';;0p4r<3met:r lqllC's dans lcs eap~ces lll'hed qUI'~ . .", at1l . j:.. 45,
ues ",tll~s ables y[1er-'
0-41 ! 193
"11 d' hYf"(l'r~Llrf 5
5ph~rLqu~ß ~ 5 cL j
1 i ~'. Tuo um,. 1'1 , !HU' J.8 /I, • ~-22 pg40)
lmgs,:/röJlen qlel t:hun'3l>deof j l1ilu:tBr
1-"",t-'..-Fn,JlTn Li] f311.- l'Jk«n CBfl 1'1 ieffiann5chl:t ,., olnnlg ffi 1 ti"koJ lell
'" a loh . N'~l:"hr . 311 , 1 lJ- 1110 r1 ·)ti a1
Ei l"O,"'Bki,P. JacobJ UeIds, l.otdoU.y qCodl'~lC fol!i\lj"'na,
"<I geooe!l1 C eH fferf;!nl;;!'nl forlll~. RonulliJ i;.t: '.\ .:Jl.h. 1, I 'j"
1')~ (1n!!)
7 rU51" D~ , ]~archer I H . r tcrdalgrb~un und
nllUEI 16.0 n, in I't"opar<3Unll
8 1 submani folds of 10"
C'C')homc'lElne _ 1-J8(1"\711
9 huwmoller.D. Fibre bumilas, I,' c GrilW' I:lill, New 'iör~~ 196,6
10 Le.. L-Civit:<l. I [,JS5-]62(1937}
11 'lfinzner ,n. F. Hyperf] !\Cih'JIi in Spl1llr.e11 I,
to appe..
12 P<lct 11. preprl",
13 No zu, K. SO~.II cosult:s ~O~~ IJ[ l :.mpolriJ
met c f olltlil te3 hyper~ll filceE. ull.ltJn.er.M ~th. SO-t: .1!, 1184-1188(1973/4)
omiz.u,K. EU l\n' s work: da hYJ?[!t"s~",
f'leus. [)irr ,Goom., symp. h. Z7 1" 5tal1r~rd 1971
15 Ollele L , H., 'i''lk ,,:uc::h i ,~I. On lIome types o{ illO~"rameulc
hyper~~;!nces in GplereS I, hoku Mi:lll\.J.~,S15-5S~.1975)
-, part II. Tohakll' ,o,ttl. ,1. ,28, 7-~S (1 971i)
1
17 :'iu-Jr13,Bo. Rend.Acad.Linc/}j :n, 203-207 (1938)
18 T<I)I;aqi,R .. Tilkatw~hi,T. On thc principa1 curvatures of
hOn'.tclgoncous hyp<>r:;ur(aces in il :)phere, Di fr<-t'entiil1
Geometry, in honer of K. Yano, killukuniya, Tokyo, 1972,
469-481
111 rot< Feru5
Fachhürcich 'tathem.a ti k
Techrd "che Universität I1fJrltn
~t"a{l des 17 _:lIJrl ~
1000 -r1in 12